Finding the quantum thermoelectric with maximal efficiency and minimal entropy production at given power output
We investigate the nonlinear scattering theory for quantum systems with strong Seebeck and Peltier effects, and consider their use as heat-engines and refrigerators with finite power outputs. This article gives detailed derivations of the results summarized in Phys. Rev. Lett. 112, 130601 (2014). It shows how to use the scattering theory to find (i) the quantum thermoelectric with maximum possible power output, and (ii) the quantum thermoelectric with maximum efficiency at given power output. The latter corresponds to a minimal entropy production at that power output. These quantities are of quantum origin since they depend on system size over electronic wavelength, and so have no analogue in classical thermodynamics. The maximal efficiency coincides with Carnot efficiency at zero power output, but decreases with increasing power output. This gives a fundamental lower bound on entropy production, which means that reversibility (in the thermodynamic sense) is impossible for finite power output. The suppression of efficiency by (nonlinear) phonon and photon effects is addressed in detail; when these effects are strong, maximum efficiency coincides with maximum power. Finally, we show in particular limits (typically without magnetic fields) that relaxation within the quantum system does not allow the system to exceed the bounds derived for relaxation-free systems, however, a general proof of this remains elusive.
pacs:73.63.-b, 05.70.Ln, 72.15.Jf, 84.60.Rb
Thermoelectric effects in nanostructures (1); (2); (3); (4) and molecules (5); (6) are of great current interest. They might enable efficient electricity generation and refrigeration (7); (8); (9), and could also lead to new types of sub-Kelvin refrigeration, cooling electrons in solid-state samples to lower temperatures than with conventional cryostats (1), or cooling fermionic atomic gases (10); (11); (12). However, they are also extremely interesting at the level of fundamental physics, since they allow one to construct the simplest possible quantum machine that converts heat flows into useful work (electrical power in this case) or vice versa. This makes them an ideal case study for quantum thermodynamics, i.e. the thermodynamics of quantum systems (13).
The simplest heat-engine is a thermocouple circuit, as shown in Fig. 1. It consists of a pair of thermoelectrics with opposite thermoelectric responses (filled and open circles) and a load, connected in a ring. Between each such circuit element is a big reservoir of electrons, the reservoir on the left () is hotter than the others, , so heat flows from left to right. One thermoelectric’s response causes an electric current to flow in the opposite direction to the heat flow (filled circle), while the other’s causes an electric current to flow in the same direction as the heat flow (open circle). Thus, the two thermoelectrics turn heat energy into electrical work; a current flow through the load. The load is assumed to be a device that turns the electrical work into some other form of work; it could be a battery-charger (turning electrical work into chemical work) or a motor (turning electrical work into mechanical work).
The same thermocouple circuit can be made into a refrigerator simply by replacing the load with a power supply. The power supply does work to establish the current around the circuit, and this current through the thermoelectrics can “drag” heat out of reservoir . In other words, the electrical current and heat flow are the same as for the heat-engine, but now the former causes the latter rather than vice versa. Thus, the refrigerator cools reservoir , so .
The laws of classical thermodynamics inform us that entropy production can never be negative, and maximal efficiency occurs when a system operates reversibly (zero entropy production). Thus, it places fundamental bounds on heat-engine and refrigerator efficiencies, known as Carnot efficiencies. In both cases, the efficiency is defined as the power output divided by the power input. For the heat-engine, the power input is the heat current out of the hotter reservoir (reservoir ), , and the power output is the electrical power generated . Thus, the heat-engine (eng) efficiency is
This efficiency can never exceed Carnot’s limit,
where we recall that we have .
For the refrigerator the situation is reversed, the load is replaced by a power supply, and the power input is the electrical power that the circuit absorbs from the power supply, . The power output is the heat current out of the colder reservoir (reservoir ), . This is called the cooling power, because it is the rate at which the circuit removes heat energy from reservoir . Thus, the refrigerator (fri) efficiency is
This efficiency is often called the coefficient of performance or COP. This efficiency can never exceed Carnot’s limit,
where we recall that (opposite of heat-engine).
Strangely, the laws of classical thermodynamics do not appear to place a fundamental bound on the power output associated with reversible (Carnot efficient) operation. Most textbooks say that reversibility requires “small” power output, but rarely define what “small” means. The central objective of Ref. [(14)] was to find the meaning of “small”, and find a fundamental upper bound on the efficiency of an irreversible system in which the power output was not small.
Ref. [(14)] did this for the class of quantum thermoelectrics that are well modelled by a scattering theory, which enables one to straightforwardly treat quantum and thermodynamic effects on an equal footing. It summarized two principal results absent from classical thermodynamics. Firstly, there is a quantum bound (qb) on the power output, and no quantum system can exceed this bound (open circles in Fig. 2). Secondly, there is a upper bound on the efficiency at any given power output less than this bound (thick black curves in Fig. 2). The efficiency at given power output can only reach Carnot efficiency when the power output is very small compared to the quantum bound on power output. The upper bound on efficiency then decays monotonically as one increases the power output towards the quantum bound. The objective of this article is to explain in detail the methods used to derive these results, along with the other results that were summarized in Ref. [(14)].
i.1 Contents of this article
This article provides detailed derivations of the results in Ref. [(14)]. The first part of this article is an extended introduction. Section II is a short review of the relevant literature. Section III discusses how we define temperature, heat and entropy. Section IV recalls the connection between efficiency and entropy production in any thermodynamic machine. Section V reviews the nonlinear scattering theory, which section VII uses to make very simple over-estimates of a quantum system’s maximum power output.
The second part of this article considers how to optimize a system which is free of relaxation and has no phonons or photons. Section VIII gives a hand-waving explanation of the optimal heat engine, while Section IX gives the full derivation. Section X gives a hand-waving explanation of the optimal refrigerator, while Section XI gives the full derivation. Section XII proposes a system which could in principle come arbitrarily close to the optimal properties given in sections IX and XI. Section XIII considers many quantum thermoelectrics in parallel.
Ii Comments on existing literature
There is much interest in using thermoelectric effects to cool fermionic atomic gases (10); (11); (12), which are hard to cool via other methods. This physics is extremely similar to that in this work, but there is a crucial difference. For the electronic systems that we consider, we can assume the temperatures to be much less than the reservoir’s Fermi energy, and so take all electrons to have the same Fermi wavelength. In contrast, fermionic atomic gases have temperatures of order the Fermi energy, so the high-energy particles in a reservoir have a different wavelength from the low-energy ones. Thus, our results do not apply to atomic gases, although our methodology does(12).
ii.1 Nonlinear systems and the figure of merit
Engineers commonly state that wide-ranging applications for thermoelectrics would require them to have a dimensionless figure of merit, , greater than three. This dimensionless figure of merit is a dimensionless combination of the linear-response coefficients (7) , for temperature , Seebeck coefficient , electrical conductance , and thermal conductance . Yet for us, is just a way to characterize the efficiency, via
with a similar relationship for refrigerators. Thus, someone asking for a device with a , actually requires one with an efficiency of more than one third of Carnot efficiency. This is crucial, because the efficiency is a physical quantity in linear and nonlinear situations, while is only meaningful in the linear-response regime (15); (16); (17); (18); (19); (20).
Linear-response theory rarely fails for bulk semiconductors, even when and are very different. Yet it is completely inadequate for the quantum systems that we consider here. Linear-response theory requires the temperature drop on the scale of the electron relaxation length (distance travelled before thermalizing) to be much less than the average temperature. For a typical millimetre-thick bulk thermoelectric between a diesel motor’s exhaust system (K) and its surroundings (K), the relaxation length (inelastic scattering length) is of order the mean free path; typically 1-100nm. The temperature drop on this scale is tens of thousands of times smaller than the temperature drop across the whole thermoelectric. This is absolutely tiny compared with the average temperature, so linear-response (21) works well, even though is of order one.
In contrast, for quantum systems (), the whole temperature drop occurs on the scale of a few nanometres or less, and so linear-response theory is inapplicable whenever is not small.
ii.2 Carnot efficiency
A system must be reversible (create no entropy) to have Carnot efficiency; proposals exist to achieve this in bulk (21) or quantum (22); (23); (24) thermoelectric. It requires that electrons only pass between reservoirs L and R at the energy where the occupation probabilities are identical in the two reservoirs (22). Thus, a thermoelectric requires two things to be reversible. Firstly, it must have a -function-like transmission (21); (22); (23); (24); (25), which only lets electrons through at energy . Secondly,(22) the load’s resistance must be such that , so the reservoirs’ occupations are equal at , see Fig. 4.
By definition this means the current vanishes, and thus so does the power output, . However, one can see how vanishes by considering a quantum system which lets electrons through in a tiny energy window from to , see Fig 5. When we take , one has Carnot efficiency, however we will see (leading order term in Eq. (48)) that
which vanishes as .
ii.3 Heat-engine efficiency at finite power output and Curzon-Ahlborn efficiency
To increase the power output beyond that of a reversible system, one has to consider irreversible machines which generate a finite amount of entropy per unit of work generated. Curzon and Ahlborn(26) popularized the idea of studying the efficiency of a heat-engine running at its maximum power output. For classical pumps, this efficiency is , which is now called the Curzon-Ahlborn efficiency, although already given in Refs. [(27); (28); (29)]. As refrigerators, these pumps have an efficiency at maximum cooling power of zero, although Refs. [(30); (31); (32); (33)] discuss ways around this.
The response of a given heat-engine is typically a “loop” of efficiency versus power (see Fig. 2) as one varies the load on the system(2). For a peaked transmission function with width (see e.g. Fig. 5), the loop moves to the left as one reduces . In the limit , the whole loop is squashed onto the axis. In linear-response language, this machine has . In this limit, the efficiency at maximum power can be very close to that of Curzon and Ahlborn (34) (the star in Fig. 2), just as its maximum efficiency can be that of Carnot(22) (see previous section). However, its maximum power output is for small (where is finite, chosen to ensure maximum power), which vanishes for , although it is much larger than Eq. (5). Fig. 2 shows that a system with larger (such as the red curve) operating near its maximum efficiency will have both higher efficiency and higher power output than the one with small (left most grey curve) operating at maximum power.
This article shows how to derive the thick black curve in Fig. 2, thereby showing that there is a fundamental trade-off between efficiency and power output in optimal thermodynamic machines made from thermoelectrics (35). As such, our work overturns the idea that maximizing efficiency at maximum power is the best route to machines with both high efficiency and high power. It also overturns the idea that systems with the narrowest transmission distributions (the largest in linear-response) are automatically the best thermoelectrics.
At this point we mention that other works(36); (38); (37); (39) have studied efficiencies for various systems with finite width transmission functions, for which power outputs can be finite. In particular, Ref. [(39)] considered a boxcar transmission function, which is the form of transmission function that we have shown can be made optimal (14).
ii.4 Pendry’s quantum bound on heat-flow
An essential ingredient in this work is Pendry’s upper bound (40) on the heat-flow through a quantum system between two reservoirs of fermions. He found this bound using a scattering theory of the type discussed in Section V below. It is a concrete example of a general principle due to Bekenstein (41), and the same bound applies in the presence of thermoelectric effects (42). The bound on the heat flow out of reservoir is achieved when all the electrons and holes arriving at the quantum system from reservoir escape into reservoir without impediment, while there is no back-flow of electrons or holes from reservoir to L. The easiest way to achieve this is to couple reservoir through the quantum system to a reservoir at zero temperature, and then ensure the quantum system does not reflect any particles. In this case the heat current equals
where is the number of transverse modes in the quantum system. We refer to this as the quantum bound (qb) on heat flow, because it depends on the quantum wave nature of the electrons; it depends on , which is given by the cross-sectional area of the quantum system divided by , where is the electron’s Fermi wavelength. As such is ill-defined within classical thermodynamics.
Iii Uniquely defining temperature, heat and entropy
Works on classical thermodynamics have shown that the definition of heat and entropy flows can be fraught with difficulties. For example, the rate of change of entropy cannot always be uniquely defined in classical continuum thermodynamics(43); (44); (45). Here the situation is even more difficult, since the electrons within the quantum systems (circles in Fig. 1) are not at equilibrium, and so their temperature cannot be defined. Thus, it is crucial to specify the logic which leads to our definitions of temperature, heat flow and entropy flow.
Our definition of heat flow originated in Refs. [(46); (47); (48)], the rate of change of entropy is then found using the Clausius relation (49) (see below). To explain these quantities and show they are unambiguous, we consider the following three step procedure for a heat engine. An analogue procedure works for a refrigerator.
Step 1. Reservoir is initially decoupled from the rest of the circuit (see Fig. 3a), has internal heat energy , and is in internal equilibrium at temperature . The rest of the circuit is in equilibrium at temperature with internal heat energy . The internal heat energy is the total energy of the reservoir’s electron gas minus the energy which that gas would have in its ground-state. As such, the internal energy can be written as a sum over electrons and holes, with an electron at energy above the reservoir’s chemical potential (or a hole at energy below that chemical potential) contributing to this internal heat energy. The initial entropies are then for .
Step 2. We connect reservoir to the rest of the circuit ( (see Fig. 3b) and leave it connected for a long time . While we assume is long, we also assume that the reservoirs are all large enough that the energy distributions within them change very little during time . Upon connecting the circuit elements, we assume a transient response during a time , after which the circuit achieves a steady-state. We ensure that , so the physics is dominated by this steady-state. Even then the flow will be noisy (50) due to the fact electrons are discrete with probabilistic dynamics. So we also ensure that is much longer than the noise correlation time, so that the noise in the currents is negligible compared to the average currents.
Step 3. After the time , we disconnect reservoir from the rest of the circuit. Again, there will be a transient response, however we assume that a weak relaxation mechanism within the reservoirs will cause the two parts of the circuit to each relax to internal equilibrium (see Fig. 3a). After this one can unambiguously identify the temperature, , internal energy and Clausius entropy of the two parts of the circuit (for ). Since the reservoirs are large, we assume .
Thus, we can unambiguously say that the heat-current out of reservoir averaged over the time is
For the above thermocouple, we treat the currents for each thermoelectric separately, writing the heat current out of reservoir as , where is the heat current from reservoir into the lower thermoelectric in Fig. 1 (the filled circle), and is the heat current from reservoir into the upper thermoelectric in Fig. 1 (the open circle). Treating each thermoelectric separately is convenient, and also allows one to generalize the results to “thermopiles”, which contain hundreds of thermoelectrics arranged so that they are electrically in series, but thermally in parallel.
The average rate of change of entropy in the circuit is , where is the average rate of change of entropy associated with the lower thermoelectric in Fig. (1), while is that for the upper thermoelectric. Then
while is the same with replaced by . We neglect the entropy of the thermoelectrics and load, by assuming their initial and final state are the same. This will be the case if they are small compared to the reservoirs, so their initial and final states a simply given by the temperature .
The nonlinear scattering theory in Ref. [(51)] captures long-time average currents (usually called the DC response in electronics), such as electrical current and heat current , see references in Section V. It is believed to be exact for non-interacting particles, and also applies when interactions can be treated in a mean-field approximation (see again section V). A crucial aspect of the scattering theory is that we do not need to describe the non-equilibrium state of the quantum system during step 2. Instead, we need that quantum system’s transmission function, defined in section V.
In this article we will only discuss the long-time average of the rates of flows (not the noisy instantaneous flows), and thus will not explicitly indicate the average; so , and should be interpreted as , and .
Iv Entropy production
There are little known universal relations between efficiency, power and and entropy production, which follow trivially from the laws of thermodynamics (52). Consider the lower thermoelectric in Fig. 1a (filled circle), with and being steady-state heat currents into it from reservoir and R. Then the first law of thermodynamics is
where is the electrical power generated. The Clausius relation for the rate of change of total entropy averaged over long times as in Eq. (8), is
where we have used Eq. (9) to eliminate .
For a heat engine, we take to be positive, which means and is negative. We use Eq. (1) to replace with in Eq. (10). Then, the rate of entropy production by a heat-engine with efficiency at power output is
where the Carnot efficiency, , is given in Eq. (2). Hence, knowing the efficiency at power , tells us the entropy production at that power. Maximizing the former minimizes the latter.
For refrigeration, the load in Fig. 1 is replaced by a power supply, the thermoelectric thus absorbs a power to extract heat from the cold reservoir. We take reservoir as cold () , so is positive. We replace by in Eqs. (9,10). We then use Eq. (3) to replace by . Then the rate of entropy production by a refrigerator at cooling power is
where the Carnot efficiency, , is given in Eq. (4). Hence knowing a refrigerator’s efficiency at cooling power gives us its entropy production, and we see that maximizing the former minimizes the latter.
Eqs. (11,12) hold for systems modelled by scattering theory, because this theory satisfies the laws of thermodynamics (53)(42). The rate of entropy production is zero when the efficiency is that of Carnot, but becomes increasingly positive as the efficiency reduces. In this article, we calculate the maximum efficiency for given power output, and then use Eqs. (11,12) to get the minimum rate of entropy production at that power output.
V Nonlinear Scattering Theory
This work uses Christen and Büttiker’s nonlinear scattering theory (51), which treats electron-electron interactions as mean-field charging effects. Refs. [(54); (17); (18)] added thermoelectric effects by following works on linear-response (46); (47); (48). Particle and heat flows are given by the transmission function, , for electrons to go from left () to right () at energy , where is a self-consistently determined function of , and . In short, this self-consistency condition originates from the fact that electrons injected from the leads change the charge distribution in the quantum system, which in turn changes the behaviour of those injected electrons (via electron-electron interactions). The transmission function can be determined self-consistently with the charge distribution, if the latter is treated in a time-independent mean-field manner (neglecting single electron effects). We note that the same nonlinear scattering theory was also derived for resonant level models (22); (36) using functional RG to treat single-electron charging effects (37).
The scattering theory for the heat current is based on the observation that an electron leaving reservoir at energy is carrying heat out of that reservoir (48), where is the reservoir’s chemical potential. Thus, a reservoir is cooled by removing an electron above the Fermi surface, but heated by removing a electron below the Fermi surface. It is convenient to treat empty states below a reference chemical potential (which we define as ), as “holes”. Then we do not need to keep track of a full Fermi sea of electrons, but only the holes in that Fermi sea. Then the heat-currents out of reservoirs L and R and into the quantum system are
where is the electron charge (), so is the chemical potential of reservoir measured from the reference chemical potential (). The sum is over for “electron” states (full states above the reference chemical potential), and for “hole” states (empty states below that chemical potential). The Fermi function for particles entering from reservoir , is
The transmission function, , is the probability that a particle with energy entering the quantum system from reservoir will exit into reservoir as a particle with energy . We only allow here, since we do not consider electron to hole scattering within the quantum system (only common when superconductors are present). Interactions mean that , is a self-consistently determined function of , and .
The system generates power , so
It is easy to verify that Eqs. (13-LABEL:Eq:Pgen) satisfy the first law of thermodynamics, Eq. (9). This theory assumes the quantum system to be relaxation-free, although decoherence is allowed as it does not change the structure of Eqs. (13-LABEL:Eq:Pgen). Relaxation is discussed in Section XV.
We define the voltage drop as . Without loss of generality we take the reference chemical potential to be that of reservoir , so
then and coincide with Eqs. (8,9) in Ref. [(14)].
Numerous works have found the properties of thermoelectric systems from their transmission functions, . Linear-response examples include Refs. [(46); (47); (48); (55); (5); (56); (57); (58); (59); (61); (60); (62); (63); (64); (65); (66)], while nonlinear responses were considered in Refs. [(67); (68); (36); (54); (17); (18); (37); (69); (70); (71)], see Refs. [(2); (3); (4)] for recent reviews. However, here we do not ask what is the efficiency of a given system, we ask what is the system that would achieve the highest efficiency, and what is this efficiency? This is similar in spirit to Ref. [(21)], except that we maximize the efficiency for given power output.
We need to answer this question in the context of the mean-field treatment of electron-electron interactions(51), in which the transmission function for any given system is the solution of the above mentioned self-consistency procedure. Despite this complexity, any transmission function (including all mean-field interactions) must obey
where is the number of transverse modes at the narrowest point in the nanostructure, see Fig. 1. Let us assume that this is the only constraint on the transmission function. Let us assume that for any given , and , a clever physicist could engineer any desired transmission function, so long as it obeys Eq. (18). Presumably they could do this either by solving the self-consistency equations for , or by experimental trial and error. Thus, in this work, we find the which maximizes the efficiency given solely the constraint in Eq. (18), and get this maximum efficiency. We then rely on future physicists to find a way to construct a system with this (although some hints are given in Section XII).
Vi From thermoelectric optimization to thermocouple optimization
The rest of this article considers optimizing a single thermoelectric. However, an optimal thermocouple heat engine (or refrigerator) consists of two systems with opposite thermoelectric responses (full and open circles in Fig. 1). So here we explain how to get the optimal thermocouple from the optimal thermoelectric.
Suppose the optimal system between and (the full circle) has a given transmission function , which we will find in Section IX. This system generates an electron flow parallel to heat flow (so electric current is anti-parallel to heat flow, implying a negative Peltier coefficient). The system between and (the open circle) must have the opposite response. For this we interchange the role played by electrons and holes compared with , so the optimal system between and has
If the optimal bias for the system between and is (which we will also find in Section IX), then the optimal bias for the system between and is . Then the heat flow from reservoir into equals that from into , while the electrical current from into is opposite to that from into , and so is the same for each thermoelectric. The load across the thermocouple (the two thermoelectrics) must be chosen such that the bias across the thermocouple is . The condition that the charge current out of equals that into will then ensure that both thermoelectrics are at their optimal bias.
In the rest of this article we discuss power output and heat input per thermoelectric. For a thermocouple, one simply needs to multiply these by two, so the efficiency is unchanged but the power output is doubled.
Vii Simple estimate of bounds on power output
One of the principal results of Ref. [(14)] is the quantum bounds on the power output of heat-engines and refrigerators. The exact derivation of these bounds is given in Sections IX.1 and XI.1. Here, we give simple arguments for their basic form based on Pendry’s limit of heat flow discussed in Section II.4 above.
For a refrigerator, it is natural to argue that the upper bound on cooling power will be closely related to Pendry’s bound, Eq. (6). We will show in Section XI.1 that this is the case. A two-lead thermoelectric can extract as much as half of . In other words, the cooling power of any refrigerator must obey
Now let us turn to a heat-engine operating between a hot reservoir and cold reservoir . Following Pendry’s logic, we can expect that the heat current into the quantum system from reservoir cannot be more than . Similarly, no heat engine can exceed Carnot’s efficiency, Eq. (2). Thus, we can safely assume any system’s power output is less than
We know this is a significant over-estimate, because maximal heat flow cannot coincide with Carnot efficiency. Maximum heat flow requires is maximal for all and , while Carnot efficiency requires a with a -function-like dependence on (see Section II.2). None the less, the full calculation in Section IX.1 shows that the true quantum bound on power output is such that (72)
where . Thus, the simple over-estimate of the bound, , differs from the true bound by a factor of . In other words it over estimates the quantum bound by a factor between 5.19 and 10.38 (that is 5.19 when and 10.38 when ). This is not bad for such a simple estimate.
Viii Guessing the optimal transmission for a heat-engine
Here we use simple arguments to guess the transmission function which will maximize a heat-engine’s efficiency for a given power output. We consider the flow of electrons from reservoir to reservoir (the filled circle Fig. 1a, remembering , so electron flow is in the opposite direction to ). To produce power, the electrical current must flow against a bias, so we require to be positive, with as in Eq. (17). Inspection of the integrand of Eq. (LABEL:Eq:Pgen) shows that it only gives positive contributions to the power output, , when . From Eq. (15), one can show that and cross at
see Fig. 4. Since is positive, we maximize the power output by blocking the transmission of those electrons () which have , and blocking the transmission all holes (). For , all energies above add to the power output. Hence, maximizing transmission for all will maximize the power output, giving . However, a detailed calculation, such as that in Section IX, is required to find the which will maximize ; remembering that depends directly on as well as indirectly (via the above choice of ).
Now we consider maximizing the efficiency at a given power output , where . Comparing the integrands in Eqs. (13,LABEL:Eq:Pgen), we see that contains an extra factor of energy compared to . As a result, the transmission of electrons () with large enhances the heat current much more than it enhances the power output. This means that the higher an electron’s is, the less efficiently it contributes to power production. Thus, one would guess that it is optimal to have an upper cut-off on transmission, , which would be just high enough to ensure the desired power output , but no higher. Then the transmission function will look like a “band-pass filter” (the “boxcar” form in Fig 5), with and further apart for higher power outputs. This guess is correct, however the choice of affects both and , so the calculation in Section IX is necessary to find the , and which maximize the efficiency for given .
Ix Maximizing heat-engine efficiency for given power output
Now we present the central calculations of this article, finding the maximum efficiency of a quantum thermoelectric with given power output. In this section we consider heat-engines, while Section XI addresses refrigerators.
For a heat engine, our objective is to find the transmission function, , and bias, , that maximize the efficiency for given power output . To do this we treat as a set of many slices each of width , see the sketch in Fig. 6a. We define as the height of the th slice, which is at energy . Our objective is to find the optimal value of for each , and optimal values of the bias, ; all under the constraint of fixed . Often such optimization problems are formidable, however this one is fairly straightforward.
The efficiency is maximum for a fixed power, , if is minimum for that . If we make an infinitesimal change of and , we note that
where indicates that the derivative is taken at constant , and the primed indicates for fixed transmission functions. If we want to fix as we change , we must change the bias to compensate. For this, we set in Eq. (25) and substitute the result for into Eq. (24). Then decreases (increasing efficiency) for an infinitesimal increase of at fixed , if
Comparing Eq. (13) and Eq. (LABEL:Eq:Pgen), one sees that
where is given in Eq. (17). Thus, the efficiency grows with a small increase of if
where , , , and are positive.
For what follows, let us define two energies
A heat-engine is only useful if , and this is only true for . Hence, if and , then is maximum for at its maximum value, . For all other and , is maximum for at its minimum value, . Since the left-hand-side of Eq. (28) is not zero for any , there are no stationary points, which is why never takes a value between its maximum and minimum values. Thus, the optimal is a “boxcar” or “top-hat” function,
see Fig. 6b. It hence acts as a band-pass filter, only allowing flow between L and R for electrons () in the energy window between to .
Substituting a boxcar transmission function with arbitrary and into Eqs. (13,LABEL:Eq:Pgen) gives
where we define
which are both positive for any . Remembering that we took and , these integrals are
for dilogarithm function, .
We are only interested in cases where fulfills the condition in Eq. (29), in this case , which means and are related to and by
Eq. (30) tells us that depends on and , but that these depend in-turn on . Hence to find , we substitutes Eqs. (32,33) into Eq. (30) to get a transcendental equation for as a function of for given . This equation is too hard to solve analytically (except in the high and low power limits, discussed in Sections IX.1 and IX.2 respectively). The red curve in Fig. 7a is a numerical solution for .
Having found as a function of for given , we can use Eqs. (32,33) to get and . We can then invert the second relation to get . At this point we can find , and then use Eq. (1) to get the quantity that we desire — the maximum efficiency at given power output, .
In Section IX.1, we do this procedure analytically for high power (), and in Section IX.2, we do this procedure analytically for low power (). For other cases, we only have a numerical solution for the transcendental equation for as a function of , so we must do everything numerically.
Fig. 8a gives the values of and which result from solving the transcendental equation numerically for a variety of different . Eq. (29) then relates to . The qualitative behaviour of the resulting boxcar transmission function is shown in Fig. 5. This numerical evaluation enables us to find the efficiency as a function of and , which we plot in Fig. 9a.
ix.1 Quantum bound on heat engine power output
Here we want to find the highest possible power output of the heat-engine. In the previous section, we had the power as a function of two independent parameters, and , with given by Eq. (29). However, we know that Eq. (30) will then determine a line in this two-dimensional parameter space (Fig. 7a), which we can parametrize by the parameter . The maximum possible power corresponds to , where we recall . This has two consequences, the first is that from Eq. (29), we see that means that . Thus, the transmission function , taking the form of a Heaviside step function, , where is given in Eq. (29). Taking Eq. (33) combined with Eq. (40) for , gives
The second consequence of , is that the -derivative of this expression must be zero. This gives us the condition that
where we define . Numerically solving this equation gives . Eq. (29) means that this corresponds to , indicated by the red arrow in Fig. 7a. Substituting this back into gives the maximum achievable value of ,
We refer to this as the quantum bound (qb) on power output(72), because of its origin in the Fermi wavelength of the electrons, . We see this in the fact that is proportional to the number of transverse modes in the quantum system, , which is given by the cross-sectional area of the quantum system divided by . This quantity has no analogue in classical thermodynamics.
The efficiency at this maximum power, , is
As such, it varies with , but is always more than . This efficiency is less than Curzon and Ahlborn’s efficiency for all (although not much less). However, the power output here is infinitely larger than the maximum power output of systems that achieve Curzon and Ahlborn’s efficiency, see Section II.3.
The form of Eq. (45) is very different from Curzon and Ahlborn’s efficiency. However, we note in passing that Eq. (45) can easily be written as , which is reminiscent of the efficiency at maximum power found for very different systems (certain classical stochastic heat-engines) in Eq. (31) of Ref. [(73)].
ix.2 Optimal heat-engine at low power output
Now we turn to the opposite limit, that of low power output,